introduction to the vc-dimension - github pagesweiningchen.github.io/notes/vc_dim.pdf · outline 1...

INTRODUCTION TO THE VC-DIMENSION

Wei-Ning Chen

December 28, 2018

WEI-NING CHEN INTRODUCTION TO THE VC-DIMENSION DECEMBER 28, 2018 1 / 40

OUTLINE

1 RECAP

2 MOTIVATION

3 THE VC-DIMENSIONDefinitionsExamples

4 THE FUNDAMENTAL THEOREM OF LEARNING THEORY

5 ADVANCED TOPICSGlivenko-Cantelli TheoremVC-entropy and Growth Function

6 EXERCISES AND DISCUSSION

DEFINITION (UNIFORM CONVERGENCE)

We say that a hypothesis class H has the uniform convergence property (w.r.t. a domainZ and a loss function `) if there exists a function mUC

H such that for every ε, δ ∈ (0, 1) andfor every probability distribution D over Z, if S is a sample of m ≥ mUC

H (ε, δ) examplesdrawn i.i.d. according to D, then

P(|LS(h)− LD(h)| ≤ ε, ∀h ∈ H) ≥ 1− δ

Equivalently,limm→∞

P(suph∈H|LS(h)− LD(h)| > ε) = 0

Remark: Compare to the difinition of PAC:

P(LD(hS)− infh∈H

LD(h) ≤ ε) ≥ 1− δ

THEOREM (NO-FREE-LUNCH)

Let A be any learning algorithm for the task of binary classification with respect to the0− 1 loss over a domain X . Let m be any number smaller than |X |/2, representing atraining set size. Then, there exists a distribution D over X × {0, 1} such that:� There exists a function f : X → {0, 1} with LD(f) = 0.

� P(LD(A(S)) ≥1

8) ≥ 1

1 RECAP

2 MOTIVATION

LEARNING FROM INFINITE-SIZE HYPOTHESIS CLASS

In chapter 2, we see that every finite hypothesis class H is learnable; moreover, thesample complexity is bounded by

mH(δ, ε) ≤log(|H|)/δ

So, what if |H| =∞?

EXAMPLE: CONCENTRIC CIRCLE

EXAMPLE

Let X = R2, Y = {0, 1}, and let H be the class of concentric circles in the plane, that is,H = {hr : r ∈ R+}. Prove that H is PAC learnable (assume realizability), and its samplecomplexity is bounded by

mH(ε, δ) ≤log(2δ)

First, we specify HB.By definition, if h ∈ HB, we have D(h(x) 6= h∗(x)) ≥ ε

h∗ h

Equivalently, D(h(x) = h∗(x)) ≤ 1− ε

h∗ h

If now H is finite, we can apply union bound:

Dm(⋃

h∈HB

∀i = [m]|h(xi) = h∗(xi)) ≤ |H|(1− ε)m ≤ δ

We can slightly modify the union bound for the case |H| =∞.

h∗hr2

Let S ∼ Dm, and rmin = minx∈S

rx, rmax = maxx∈S

We havePS∼Dm(LD(hS) ≥ ε) ≤ PS∼Dm(rmin ≥ r1 ∪ rmax ≤ r0)

≤ PS∼Dm(rmin ≥ r1) + PS∼Dm(rmax ≤ r0)≤ 2(1− ε)m ≤ 2e−mε ≤ δ

Therefore, for all ε and δ, the sample complexity can be bounded by

m ≤ log(2/δ)

� In chapter 2, we see that every finite hypothesis class H is learnable; moreover, thesample complexity is bounded by

mH(δ, ε) ≤log(|H|)/δ

� Also, we see some examples that even the class is infinite-size, it may still belearnable.

� Therefore, we need a measure of H’s complexity

� In this cahpter, we will formally define the complexity of H (VC dimension) ,and show that

H has uniform convergence property ⇐⇒ VCdim(H) <∞

1 RECAP

2 MOTIVATION

THE VC-DIMENSION

DEFINITION (RESTRICTION H TO C)

Let H be a class of function from X to {0, 1} and let C = {c1, ..., cm} ⊂ X . The restrictionof H to C is the set off all functions from C to {0, 1} that can be derived from H. That is,

HC = {h(c1), ..., h(cm)) : h ∈ H

DEFINITION (SHATTERING)

A hypothesis class H shatters a finite set C if the restriction of H to C is the set of allfunctions from C to {0, 1}. That is,|HC | = 2|C|.

THE VC-DIMENSION

COROLLARY (COROLLARY6.4)

Let H be a hypothesis class of functions from X to {0, 1}. Let m be a training set size.Assume that there exists a set C of size 2m that is shattered by H.Then, for any learning algorithm A

P(LD(A(S)) ≥1

8) ≥ 1

Remark: This is a direct result from NFL Theorem

Remark2: If for all m, there exists a set C of size 2m that is shattered by H, then H is notPAC learnable

THE VC-DIMENSION

DEFINITION (VC-DIMENSION)

The VC-dimension of a hypothesis class H, denoted VCdim(H), is the maximal size of aset C ⊂ X that can be shattered by H. If H can shatter sets of arbitrarily large size, wesay that H has infinite VC-dimension.

Remark: If VCdim(H)=d, it means that

∃C ⊂ X that can be shattered by H,

NOT∀C ⊂ X that can be shattered by H,

EXAMPLE: THRESHOLD FUNCTIONS

� Let H be the all threshold function on R.

� For an arbitrary set C = {c1}, H shatters C,therefore VDdim(H) ≥ 1.

� For an arbitrary set C = {c1, c2} , H does not shatter C.Therefore, VDdim(H) < 2.

EXAMPLE: INTERVALS

� Let H be the intervals over R; that is, H = {1[a,b](x)|a, b ∈ R}

� It is easy to show that VCdim(H) = 2

EXAMPLE: AXIS ALIGNED RECTANGLES

� Let H be the the class of axis aligned rectangles:

H = {1[a,b]×[c,d]|a, b, c, d ∈ R}

� VDdim(H) = 4

EXAMPLE: AXIS ALIGNED RECTANGLES

1 RECAP

2 MOTIVATION

THE FUNDAMENTAL THEOREM OF LEARNING THEORY

THEOREM (THE FUNDAMENTAL THEOREM OF STATISTICAL LEARNING)

Let H be a hypothesis class of functions from a domain X to {0, 1} and let the lossfunction be the 0-1 loss. Then, the following are quivalent:

1 H has the uniform covergence property.2 Any ERM rule is a successful agnostic PAC learner for H.3 H is agnostic PAC learnable.4 H is PAC learnable.5 Any ERM rule is a successful PAC learner for H.6 H has a finite VC-dimension.

GROWTH FUNCTION ANS SAUER’S LEMMA

The growth function measures the maximal ”effective? size of H on a set of m examples.

DEFINITION (GROWTH FUNCTION)

Let H be a hypothesis class. Then the growth function of H is defined as

τH(m) = supC⊂X :|C|=m

In words, τH(m) is the number of different functions from a set C of size m to {0, 1} thatcan be obtained by restricting H to C.

Remark: if VCdim(H) = d, then for any m ≤ d we have τH(m) = 2m.However, what interesting is the case m ≥ d.

GROWTH FUNCTION AND SAUER’S LEMMA

THEOREM (SAUER’S LEMMA)

Let H be a hypothesis with VCdim(H) = d. Then for all m,

τH(m) ≤d∑i=0

)≤ (em/d)d

Remark: if VCdim(H) is finite, then the growth function is polynomial in m.

UNIFORM CONVERGENCE IN VC CLASS

THEOREM (UNIFORM CONVERGES IN VC CLASS (THEOREM 6.11))

Let H be a class and let τH(m) be its growth function. Then, for every D and every δ

PS∼Dm(|LD(h)− LS(h)| > ε) ≤ δ

where ε can be choose as4 +

√(log(τH(2m)))

δ√2m

. In other words, this theorem tells us that

VCdim(H) <∞⇐⇒ lim`→∞

log τH(`)

`= 0⇐⇒ uniform convergence property holds.

THE FUNDAMENTAL THEOREM OF LEARNING THEORY

THEOREM (THE FUNDAMENTAL THEOREM-QUANTITATIVE VERSION)

Let H be a hypothesis class from a domain X to {0, 1} and let the loss function be the 0-1loss. Then, there are absolute constants C1, C2 such that:

1 H has the uniform covergence property with sample complexity

C1d + log(1/δ)

ε2≤ m

UCH ≤ C2

d + log(1/δ)

2 H is agnostic PAC learnable with sample complexity

C1d + log(1/δ)

ε2≤ mH ≤ C2

d + log(1/δ)

3 H is PAC learnable with sample complexity

C1d + log(1/δ)

ε≤ mH ≤ C2

d log(1/ε) + log(1/δ)

1 RECAP

2 MOTIVATION

GLIVENKO-CANTELLI THEOREM

DEFINITION (EMPIRICAL DISTRIBUTION)

Let X1, ..., Xn be i.i.d. random variables in R with common cdf F (x). The empiricaldistribution function for X1, ..., Xn is given by

Fn(x) =1

1(−∞,x](Xi)

THEOREM (GLIVENKO-CANTELLI THEOREM)

‖Fn − F‖∞ = supx∈R|Fn(x)− F (x)| → 0 almost surely

Remark: ∀x, Fn(x)→ F (x) trivially by LLNWEI-NING CHEN INTRODUCTION TO THE VC-DIMENSION DECEMBER 28, 2018 30 / 40

GLIVENKO-CANTELLI THEOREM

� More generally, consider a space X and σ−field F generated by borel set withprobability measure P and empirical measure Pn

� Then F (x) = P ((−∞, x]), and Fn(x) = Pn((−∞, x])

� We can rewrite supx∈R|Fn(x)− F (x)| as sup

c∈C|Pn(C)− P (C)|,

where C = {(−∞, x]|x ∈ R}

� What happens for the general C ?

GLIVENKO-CANTELLI CLASS

DEFINITION (GC-CLASS)

Let C ⊂ {C|C measurable in X}. Then if

‖Pn − P‖C = supc∈C|Pn(C)− P (C)|

we say class C is a Glivenko-Cantelli class.

DEFINITION (UNIFORMLY GC)

A class is called uniformly Glivenko-Cantelli if the convergence occurs uniformly over allprobability measures P on (X ,F):

supP∈P(S,A)

E‖Pn − P‖C → 0

VC CLASS

DEFINITION (VC CLASS)

A class with finite VC dimension is called a Vapnik-Chervonenkis class or VC class

THEOREM (VAPNIK AND CHERVONENKIS, 1968)

A class of sets Cis uniformly GC if and only if it is a Vapnik-Chervonenkis class

1 RECAP

2 MOTIVATION

VC ENTROPY AND GROWTH FUNCTION

In previous lecture, we define the growth function as

τH(m) = supC⊂X :|C|=m

Let’s rewrite the growth function as another form:

DEFINITION (VAPNIK)

Let x1, ...xm be m samples from X . Then define the number

NH(x1, ..., xm) = | {h(x1), ..., h(xm)|h ∈ H} | = |H{x1,...,xm}|

ObviouslyτH(m) = sup

C⊂X :|C|=m|HC | = sup

{x1,...,xm}⊂HNH(x1, ..., xm)

The supremum is taken so that the bound works even in the worst distribution.In general case, we can replace supremum by exapectation w.r.t. a specific distribution,which gives another value:

DEFINITION (VC-ENTROPY, ANNEALED VC-ENTROPY, GROWTH FUNCTION)

Let NH(x1, ..., xm) be defined as previous. Then we defined VC-entropy as

HH(m) = E logNH(x1, ..., xm)

The annealed VC-entropy as

HHann(m) = logENH(x1, ..., xm)

And the growth function (with logarithm) as

GH(m) = log supx1,...,xm

NH(x1, ..., xm)(= log τH(m))

COROLLARY

HH(m) ≤ HHann(m) ≤ GH(m)

Rmark: The fundamental theorem of learning theory tells us

limm→∞

m= lim

m→∞

log τH(m)

m= 0⇐⇒ lim

m→∞P(sup

h∈H|LS(h)− LD(h)| > ε) = 0

The convergence is uniform for all distribution P in (X ,F)

THREE MILESTONES OF LEARNING THEORY

THEOREM (VAPNIK)

limm→∞

D(suph∈H|LS(h)− LD(h)| > ε) = 0

is sufficient and necessary that

limm→∞

D(suph∈H|LS(h)− LD(h)| > ε) ≤ e−cε2m ( fast decay )

is sufficient if

limm→∞

HHann(m)

THREE MILESTONES OF LEARNING THEORY

THEOREM (VAPNIK)

limm→∞

P (suph∈H|LS(h)− LD(h)| > ε) = 0 ,for all P ∈ (X ,F)

is sufficient and necessary that

limm→∞

1 RECAP

2 MOTIVATION

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